Related papers: Random Graphs and the Parity Quantifier

We view hyper-graphs as incidence graphs, i.e. bipartite graphs with a set of nodes representing vertices and a set of nodes representing hyper-edges, with two nodes being adjacent if the corresponding vertex belongs to the corresponding…

We study the problem of computing the parity of the number of homomorphisms from an input graph $G$ to a fixed graph $H$. Faben and Jerrum [ToC'15] introduced an explicit criterion on the graph $H$ and conjectured that, if satisfied, the…

In this paper we find an integer $h=h(n)$ such that the minimum number of variables of a first order sentence that distinguishes between two independent uniformly distributed random graphs of size $n$ with the asymptotically largest…

Since Tinhofer proposed the MinGreedy algorithm for maximum cardinality matching in 1984, several experimental studies found the randomized algorithm to perform excellently for various classes of random graphs and benchmark instances. In…

We study first-order model checking, by which we refer to the problem of deciding whether or not a given first-order sentence is satisfied by a given finite structure. In particular, we aim to understand on which sets of sentences this…

We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n tend to infinity. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the…

We consider an evolving preferential attachment random graph model where at discrete times a new node is attached to an old node, selected with probability proportional to a superlinear function of its degree. For such schemes, it is known…

A countable, bounded degree graph is almost finite if it has a tiling with isomorphic copies of finitely many F\o lner sets, and we call it strongly almost finite, if the tiling can be randomized so that the probability that a vertex is on…

Many computational problems can be modelled as the class of all finite structures $\mathbb A$ that satisfy a fixed first-order sentence $\phi$ hereditarily, i.e., we require that every (induced) substructure of $\mathbb A$ satisfies $\phi$.…

This paper establishes model-theoretic properties of $\mathrm{FOE}^{\infty}$, a variation of monadic first-order logic that features the generalised quantifier $\exists^\infty$ (`there are infinitely many'). We provide syntactically defined…

Random planar graphs have been the subject of much recent work. Many basic properties of the standard uniform random planar graph P_{n}, by which we mean a graph chosen uniformly at random from the set of all planar graphs with vertex set…

When studying networks using random graph models, one is sometimes faced with situations where the notion of adjacency between nodes reflects multiple constraints. Traditional random graph models are insufficient to handle such situations.…

The law of a finite graph is a probability measure induced by the orbits of the graph under its automorphism group. Every law satisfies the intrinsic mass transport principle, which is also known as unimodularity. We discuss the convergence…

We consider finite relational signatures $\tau \subseteq \sigma$, a sequence of finite base $\tau$-structures $(\mathcal{B}_n : n \in \mathbb{N})$ the cardinalities of which tend to infinity and such that, for some number $\Delta$, the…

Consider the uniform random graph $G(n,M)$ with $n$ vertices and $M$ edges. Erd\H{o}s and R\'enyi (1960) conjectured that the limit $$ \lim_{n \to \infty} \Pr\{G(n,\textstyle{n\over 2}) is planar}} $$ exists and is a constant strictly…

In this paper, we disprove EMSO(FO$^2$) convergence law for the binomial random graph $G(n,p)$ for any constant probability $p$. More specifically, we prove that there exists an existential monadic second order sentence with 2 first order…

Let $G=(V,E)$ be a finite, connected graph. We investigate a notion of boundary $\partial G \subseteq V$ and argue that it is well behaved from the point of view of potential theory. This is done by proving a number of discrete analogous of…

For a given graph $G$, each partition of the vertices has a modularity score, with higher values indicating that the partition better captures community structure in $G$. The modularity $q^*(G)$ of the graph $G$ is defined to be the maximum…

In this paper we study the behaviour of the domination number of the Erd\H{o}s-R\'enyi random graph $\mathcal{G}(n,p)$. Extending a result of Wieland and Godbole we show that the domination number of $\mathcal{G}(n,p)$ is equal to one of…

The first detection of a quantum particle on a graph has been shown to depend sensitively on the sampling time {\tau} . Here we use the recently introduced quantum renewal equation to investigate the statistics of first detection on an…